Question: Simplify; express your answer in exponential form. Assume $z\neq 0, n\neq 0$. $\dfrac{{(z^{-5})^{2}}}{{z^{-2}n^{-1}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${z^{-5}}$ to the exponent ${2}$ . Now ${-5 \times 2 = -10}$ , so ${(z^{-5})^{2} = z^{-10}}$ In the denominator, we can use the distributive property of exponents. ${z^{-2}n^{-1} = z^{-2}n^{-1}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(z^{-5})^{2}}}{{z^{-2}n^{-1}}} = \dfrac{{z^{-10}}}{{z^{-2}n^{-1}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-10}}}{{z^{-2}n^{-1}}} = \dfrac{{z^{-10}}}{{z^{-2}}} \cdot \dfrac{{1}}{{n^{-1}}} = z^{{-10} - {(-2)}} \cdot n^{- {(-1)}} = z^{-8}n$.